A Number Out of Nowhere

Pi turns up in sums that have no circle in sight. Add and subtract odd fractions forever, 1 - 1/3 + 1/5 - 1/7 + ..., and the total creeps toward pi/4.

Pick two whole numbers at random. The chance they share no common factor is 6/pi-squared. Why a circle's constant decides a coin-flip about factors is one of the quiet shocks of mathematics.

Pi keeps appearing in physics, far from any drawn circle. It sits inside Einstein's equations of general relativity, in the swing time of a pendulum, in Heisenberg's uncertainty principle, and in the law for the force between two charges.

The reason is hidden rotation. Anything that swings, waves, or orbits is secretly circular motion in disguise, and pi is the toll you pay to describe it.